Solvability for a Class of Integro-Differential Inclusions Subject to Impulses on the Half-Line

Paola Rubbioni

doi:10.3390/math10020224

Abstract

In this paper, we study a semilinear integro-differential inclusion in Banach spaces, under the action of infinitely many impulses. We provide the existence of mild solutions on a half-line by means of the so-called extension-with-memory technique, which consists of breaking down the problem in an iterate sequence of non-impulsive Cauchy problems, each of them originated by a solution of the previous one. The key that allows us to employ this method is the definition of suitable auxiliary set-valued functions that imitate the original set-valued nonlinearity at any step of the problem’s iteration. As an example of application, we deduce the controllability of a population dynamics process with distributed delay and impulses. That is, we ensure the existence of a pair trajectory-control, meaning a possible evolution of a population and of a feedback control for a system that undergoes sudden changes caused by external forces and depends on its past with fading memory.

Highlights

IntroductionWe study the existence of solutions to the semilinear integro-differential inclusion t y (t) ∈ A(t)y(t) + F t, y(t), k(t, s)y(s)ds , t ≥ t0,
In this paper, we study the existence of solutions to the semilinear integro-differential inclusion t y (t) ∈ A(t)y(t) + F t, y(t), k(t, s)y(s)ds, t ≥ t0, (1)t0 with the initial condition y(t0) = v ∈ E, where E is a real Banach space
One of the main reasons for this research is that these equations are well suited to serve as a model for real phenomena such as heat transfer or the spread of epidemics or population dynamics, in which it is significant to take into account the spatial diffusion of the phenomenon or the past of the phenomenon itself (e.g., [9,10])

Summary

Introduction

We study the existence of solutions to the semilinear integro-differential inclusion t y (t) ∈ A(t)y(t) + F t, y(t), k(t, s)y(s)ds , t ≥ t0,. As far as we know, this method was first used in [28] and in [29] without memory and with functional delay, respectively, but on a compact interval in both cases It has been firstly used on the half-line in [25] to provide the existence of mild solutions of an impulsive Cauchy problem driven by the semilinear differential equation with functional delay y (t) = A(t)y(t) + f (t, y(t), yt), t ≥ t0, where yt(θ) := y(t + θ), θ ∈ [−τ, 0], t ∈ [t0, +∞[. We solve the feedback control problem by rewriting the model as an impulsive Cauchy problem driven by a semilinear integro-differential inclusion in the space E = L2([0, 1]) In this way, we can apply the general result obtained in Section 3 and lead to the existence of a pair, trajectory-control, providing the controllability of the population dynamics process

Essential Preliminary Notions

Existence of Impulsive Mild Solutions on the Half-Line

Example of Application

Conclusions